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If a and b are positive integers
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Updated on: 10 Sep 2018, 03:49
Updated on: 10 Sep 2018, 03:49
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If a and b are positive integers such that \(a-b\) and \(\frac{a}{b}\) are both even integers. Which of the following must be an odd integer?
A) \(\frac{a}{2}\)
B) \(\frac{b}{2}\)
C) \(\frac{a+b}{2}\)
D) \(\frac{a+2}{2}\)
E) \(\frac{b+2}{2}\)
A) \(\frac{a}{2}\)
B) \(\frac{b}{2}\)
C) \(\frac{a+b}{2}\)
D) \(\frac{a+2}{2}\)
E) \(\frac{b+2}{2}\)
Originally posted by AchyuthReddy on 09 Sep 2018, 21:20.
Last edited by Carcass on 10 Sep 2018, 03:49, edited 1 time in total.
Last edited by Carcass on 10 Sep 2018, 03:49, edited 1 time in total.
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Re: If a and b are positive integers
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10 Sep 2018, 08:20
10 Sep 2018, 08:20
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\(a-b\) even --> either both even or both odd
\(\frac{a}{b}\) even --> either both even or \(a\) is even and \(b\) is odd.
As both statements are true --> \(a\) and \(b\) must be even.
As \(\frac{a}{b}\) is an even integer --> \(a\) must be multiple of 4.
Options A is always even.
Options B can be even or odd.
Options C can be even or odd.
Options D: \(\frac{a+2}{2}=\frac{a}{2}+1\), as \(a\) is multiple of \(4\), \(\frac{a}{2}\) is even integer --> even+1=odd. Hence option D is always odd.
Options E can be even, odd.
Answer: D.
Hope this helps. The answer is one and only.
Regards
\(\frac{a}{b}\) even --> either both even or \(a\) is even and \(b\) is odd.
As both statements are true --> \(a\) and \(b\) must be even.
As \(\frac{a}{b}\) is an even integer --> \(a\) must be multiple of 4.
Options A is always even.
Options B can be even or odd.
Options C can be even or odd.
Options D: \(\frac{a+2}{2}=\frac{a}{2}+1\), as \(a\) is multiple of \(4\), \(\frac{a}{2}\) is even integer --> even+1=odd. Hence option D is always odd.
Options E can be even, odd.
Answer: D.
Hope this helps. The answer is one and only.
Regards
General Discussion
Re: If a and b are positive integers
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10 Sep 2018, 06:23
10 Sep 2018, 06:23
2
The answers are D and E.
Since a-b = even and a/b = even, then both a and b are positive even integers.
We can rewrite D as a/2 + 2/2. a/2 will always be even, 2/2 = 1. Even+1 = Odd.
The same for E
Since a-b = even and a/b = even, then both a and b are positive even integers.
We can rewrite D as a/2 + 2/2. a/2 will always be even, 2/2 = 1. Even+1 = Odd.
The same for E
